Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,16)(11,15,12,14)(17,18,21,23)(19,20,24,22);(6,8)(9,11,13,12)(10,14,16,15)(17,19,21,24)(18,22,23,20);(5,6)(7,8)(11,12)(14,15)(18,23)(20,22);(1,2)(3,4)(9,12,13,11)(10,15,16,14)(17,20,21,22)(18,19,23,24);(9,10)(11,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72331494, #72331496, #72331591 |
| maximal normal subgroup of | #72331494, #72331496, #72331591 |
| maximal quotient of | #72333121 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2976244);